Finite basis problem over tropical domains with negative entries or minus infinity

Determine whether full tropical matrix ai-semirings over coefficient domains allowing negative entries or the element [?] -\infty have a finite equational basis, since the cubic-padding argument does not resolve this question.

Background

The paper proves nonfinite basability for full max-plus matrix ai-semirings over nonnegative integer, rational, and real coefficient domains by using dimension-independent cubic-padding identities. The argument depends essentially on nonnegative entries: a cube of a matrix provides a uniform entrywise lower bound, which allows the padded identities to hold.

The authors exhibit failures of the padded identities when negative entries or -\infty are admitted. In particular, negative entries can reverse the required inequality, while assigning the tropical identity matrix causes the padding to disappear and reduces the identity to one that already fails for full matrices. These observations delimit the method but do not settle whether the corresponding full-matrix theories are finitely based.